Charged vacancies in ferroelectric PbTiO3: Formation energies, optimal Fermi region, and influence on local polarization

First-principles density-functional calculations are used to study the charged vacancies in ferroelectric perovskite PbTiO3. We systematically determine the vacancy formation energies of V-Pb, V-Ti, V-O1, and V-O2 as a function of the Fermi energy E-F of the electron reservoir and as a function of the chemical potentials of the atomic reservoirs. The study reveals (i) the stability of different charged states for each type of vacancy and the vacancy specie that is most likely to exist at a given Fermi energy, and (ii) the optimal Fermi energy that may simultaneously minimize the concentrations of all four types of vacancies and is thus good for the growth of PbTiO3 crystal. Our calculations further reveal two hidden rules in the defect physics of ferroelectric PbTiO3, namely, (iii) it is possible to grow high-quality PbTiO3 crystal even in an oxygen-poor environment, provided that the Fermi energy is wisely chosen, and (iv) the formation energies of two key vacancies (V-Pb(2-) and V-O(2+)) in the optimal Fermi-energy region are nearly independently of the choice of the oxygen chemical potential and are constrained to similar to 1.3 eV. Furthermore, we reveal, as well as provide microscopic insight on, how the presence of different vacancies may affect the local polarization in PbTiO3. We find that V-Pb(2-) is benign, but V-O1(2+) and V-O2(2+) are detrimental, to ferroelectricity.